Wt is Wiener process martingale. tau is a stopping time with respect to a filtration F

(4) For a > 0 let To be defined by To := inf{t > 0:W> a}. Prove that ET, = 00. (1) Assume that the price of a stock at time t is S = So exp(oW+at), where So >0,0 >0 and a are constants, and W = (W) is a Wiener martingale with respect to a filtration (Ft)>0- Prove that S is a martingale with respect to (File, if and only if a= -0. 15 marks (2) Let (X1)e>o be a continuous process and let (Ft)exo be its history, i.e., F = o(X, 0. For a ER consider the hitting time to defined by Ta := inf{t > 0: X > a}. Show that T, is a stopping time with respect to (F1)120 Hint: Use that (7 > t] = [max,s+ X, 0. Lett be a stopping time with respect to F, such that ET 0 let To be defined by T = inf{t > 0:Wi> a}. Prove that ET, = . (5 marks (4) For a > 0 let To be defined by To := inf{t > 0:W> a}. Prove that ET, = 00. (1) Assume that the price of a stock at time t is S = So exp(oW+at), where So >0,0 >0 and a are constants, and W = (W) is a Wiener martingale with respect to a filtration (Ft)>0- Prove that S is a martingale with respect to (File, if and only if a= -0. 15 marks (2) Let (X1)e>o be a continuous process and let (Ft)exo be its history, i.e., F = o(X, 0. For a ER consider the hitting time to defined by Ta := inf{t > 0: X > a}. Show that T, is a stopping time with respect to (F1)120 Hint: Use that (7 > t] = [max,s+ X, 0. Lett be a stopping time with respect to F, such that ET 0 let To be defined by T = inf{t > 0:Wi> a}. Prove that ET, = . (5 marks